A property on monochromatic copies of graphs containing a triangle

Abstract

A graph $H$ is called common and respectively, strongly common if the number of monochromatic copies of $H$ in a 2-edge-coloring $\phi$ of a large clique is asymptotically minimised by the random coloring with an equal proportion of each color and respectively, by the random coloring with the same proportion of each color as in $\phi$. A well-known theorem of Jagger, {\v S}t'ov{\' i}{\v c}ek and Thomason states that every graph containing a $K_4$ is not common. Here we prove an analogous result that every graph containing a $K_3$ and with at least four edges is not strongly common.